Gang, F.Y.

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PFC/JA-93-12Fast Wave Flow Drive in a Two-ComponentPlasma with Low Minority ConcentrationF. Y. GangJune, 1993Plasma Fusion centerMassachusetts Institute of TechnologyCambridge, MA 02139U. S. A.This work was supported by the U.S. Department of Energy under contract No. DE-FG02-91ER-54109. Reproduction, translation, publication, use, and disposal, in whole orin part, by or for the U.S. Government is permitted.Submitted for Publication to: Physics of fluids BFast Wave Flow Drive In a Two-Component Plasmawith Low Minority Ion ConcentrationF. Y. GangPlasma Fusion Center, Massachusetts Institute of TechnologyCambridge, MA 02139AbstractFast wave flow drive in a two-component plasma with low minority ion concentrationis investigated. It is found that strongly sheared poloidal flow can be generated in asmall region near the hybrid ion-ion resonance due to the rapid spatial variation of plasmaresponse. The flow drive depends strongly on local plasma parameters such as the plasmatemperature and density and is shown to be very effective in the edge region of a tokamakwhere the plasma temperature is low. The result is applied to a discussion of shear flowsuppression of edge turbulence.PACS Nos. 52.40.Db, 52.50.Gj, 52.35.Ra1Sheared poloidal flow has been found to play a crucial rule in suppressing plasmaturbulence, thereby improving tokamak confinement' 3 . Finding practical means of gen-erating and control of sheared poloidal flow thus becomes an important and challengingtask in current fusion research. Towards this end, several scenarios of flow drive usingexternally launched radio frequency (RF) waves have been proposed4 . In these scenarios,the externally launched RF waves are either the mode converted low frequency kineticAlfven wave or the high frequency ion Bernstein wave. The common features of thesescenarios are that the RF waves utilized have very short radial wavelengths comparable tothe ion Larmor radius, and the flow drive is determined mainly by the radial propagationof the waves. In this letter, we will present a different scenario of flow drive using thefast magnetosonic wave. For fast waves, the radial wavelength is much longer than theion Larmor radius. The flow drive is determined by the spatial variation of the plasmaresponse and depends strongly on local plasma parameters such as the plasma temperatureand density. In particular, the flow drive is shown to be very effective in the edge regionof a tokamak where the plasma temperature is low.We consider a two-component plasma in the low minority ion concentration regime.In such regime, the hybrid ion-ion resonance' falls into the minority ion resonance layer sothat the fast wave does not mode convert into the ion Bernstein wave. In the high minorityion concentration regime, the fast wave will mode convert into the ion Bernstein wave7 '8and the physics of flow drive will be similar to that described in Ref.5. For simplicity,we consider a slab model of a tokamak plasma. The equilibrium magnetic field B0 is in z(toroidal) direction. Its magnitude BO varies only in x (radial) direction and is inverselyproportional to the plasma major radius R. The fast wave is assumed to propagate in xdirection only. The allowed mode of propagation is determined by the dispersion relation 9 :2K 2N K ,N _K. (1)and the electric field polarization:-- =y -(2)E y K.x - NZ2where N. = k2tA/w, and N, = kvA/w are the radial and parallel component of theAlfven refractive index, k. and k, are the radial and parallel wavenumber respectively,VA = B 2 /47rp is the Alfven speed, w is the wave frequency, pTm = pM + pm is thetotal plasma mass density, pm and pm are the majority and minority ion mass densityrespectively, K,. and Ky are components of a normalized dielectric tensor. For a two-component plasma with minority ion resonance, K., and K,, are given by'0 :(O_)2 1 p. m nfl n_ _K -X = 1 --- 1 1 - ' am Z(am )] (3a)( )2 2 pm +0 O On- n(-K =+ ,, J ' n + wi aZ(m, )]SW2 _ (M2 12 pM W W + On W - 11! (3b)where f; is the ion cyclotron frequency, the superscript 'M' and 'm' refer to majority andminority ions respectively, amr = "! , vl is the minority ion thermal speed, and Z(am )is the plasma dispersion function.The physics of RF flow drive is based on the following plasma poloidal momentumbalance equation:Y) + (V Y)} = ( T ) + Y((Jx B),) - pp(v,) (4)where (V,) is the mean poloidal flow velocity, V is the fluctuating velocity, E and . arethe fast wave electric and magnetic field, -T and Jare the total plasma charge and currentdensity, p is the neoclassical poloidal damping rate"1 , and the average (- ) is takenover one fast wave oscillation period.3In Eq.(4), the first two terms on the right hand side are the electric and JxB forcedirectly associated with the fast wave electromagnetic fields, while the second term on theleft hand side is the nonlinear inertial force called Reynolds stress. From V - E = 47rand noting that E. < E. and k. ~ k,, the electric force can be expressed as:To calculate the JxB force, we express f in terms of OT and in terms of a us-ing the charge conservation and the Faraday's law respectively. Assuming {, B} Ocexp{i fx kdx' + ikz - iwt}, and noting that J4 < J, and b, < i, we have: 4h ~ ;and B, = (c/w)kEy. The JxB force can thus be written as:1 . .--((J xB), ~ -( y5) (6)Comparing Eqs.(5) and (6) shows that the JxB force has the same magnitude but oppositesign as the electric force. The total electromagnetic force which is the sum of the two thustends to vanish and have little effect on the temporal evolution of the mean poloidal flow.To calculate the Reynolds stress, we express the velocity response in terms of a mobilitytensor M: V = (c/B) M -. For a two-component plasma with one minority component,M=Mm +(pm/pm) Mm, where MM and Mm are the mobility tensor for the majority andminority ions respectively. Since the ion's motion are primarily in the plane perpendicularto the equilibrium magnetic field, the mobility tensor has the form:4- im" M.M= (iM: (7)MXY iAMIwhere M_, and My are given by1 O:4W (AY)2 1Pm OT nW2 = + { W _+* - a'iZ(ai)}(8a)_ (f24)2 o~ 2'' ___M- M +2 + aiZ(a'i)} (8b)Since IMXXI~ 1, and IM.,I~ 1, the magnitude of the Reynolds stress can be estimated:(i fi) (9 V) > (_ )2 (EV E)where the' >' sign applies when the spatial variation of M is faster than that of the E field.The ratio between the Reynolds stress and the electric force is: pm(V - Y)/(jA,) ;>c2/vA > 1. Therefore, the Reynolds stress is larger than either the electric or the J x Bforce (much larger than the two combined) and is the dominant force for flow drive! Inthe following, we will concentrate only on the Reynolds stress.Using Eq.(2), we can express V in terms of E, only: V, = (c/B)St5,, and V, =i(c/B)Sty, where SX and Sy are the ion velocity responses:- N2) + M (Ka. , = MyK-., - N ,K. (9a)S - M.,(KX - NZ +M2 KK- 2MX(K - N) + M (9b)Carrying out the temporal average, we have:(V -VYj) = -Re{VP* } (10)The spatial derivative of Vy in the above equation has two contributions:0 - c aV = i-[(F 3) + ikxSy]Ey (11)where the first term is due to the radial variation of the ion response S,, and the secondterm is due to the radial propagation of the fast wave. We note that S. changes rapidly5near K,, ~ N', the hybrid ion-ion resonance. In the low minority ion concentrationregime, this hybrid ion-ion resonance falls into the minority ion resonance layer, and we canapproximate OS,/O9x S,/6x where bx = R(k 1v!"/w) is the minority ion resonance layerwidth. If we take kl ~ 1/R, the ratio between the two terms in Eq.(11) is: (k.3S)/(L,kpi < 1. Therefore, the fast wave flow drive is determined by the spatial variation ofthe plasma response rather than the radial propagation of the waves. As a result, the flowdrive depends strongly on local plasma parameters such as temperature and density. Inparticular, the flow drive is expected to be more effective at lower plasma temperaturesbecause at lower plasma temperatures, the minority ion resonance layer is narrower andthe plasma response varies faster.To proceed, we substitute Eq.(11) into Eq.(10) and have:(~. -T = (1 *2VB-) = - ( {Im(Sw* ) + [k Re(S*Sy) - kIIm(S*S,)]}|5 ,| 2 (X) (12)where kR and k. are the real and imaginary part of k, respectively, and the spatial variationof the fast wave E field amplitude is taken into account through:IE5I 2 (x) = IEI 2(Xi) exp{-2 k (z')dx'} (13)where x1 is some initial spatial position. To simplify the algebra, we define:A = M.y(K. - N2) +M.Ky (14a)B = K, - N (14b)C = M,(K., - N) + MVK (14c)so that S. = A/B, and S. = C/B. The last two terms in Eq.(12) are given by:Re(S*S.) = 1 B {Re(A)Re(C) + Im(A)Im(C)} (15a)Im(S*S,) = 1 {Re(A)Im(C) - Im(A)Re(C)} (15b)6After lengthy but straight forward calculation, the first term in Eq.(12) is:I(* ) "e(~ )2(B-I -x R pm kv" 1B14x{[Re(A)Re(B) - Im(A)Im(B)][a"\ImZ(am I)]-[Re(A)Im(B) + Im(A)Re(B)][1 + amiReZ(am )]} (16)withB - K =" N (17a)z PM W IB - nM = (n - 1) t N2 (17b)w w W2-(n4)2 pM W W+ 2The Reynolds stress is completely determined by Eqs.(12)-(17). From Eq.(4), the steadystate mean poloidal flow is obtained by setting 8(V,)/Ot = 0:(V) = - (18)PneoTo illustrate the results, we solve Eqs.(12)-(18) numerically. To be specific, we assumea-two-component plasma with tritium majority and deuterium minority. The followingparameters relevant to edge plasma are used: R=150cm, a=50cm, BO=2T, Ti=100eV,nT = 1013 cm-3, nD/nT = 0.005, w=96.13MHz, and the parallel wavelength A = 130cm.The minority ion resonance position is chosen at R = 190cm. In figure 1, we showthe spatial profile of the flow velocity (V)(x). More preciously, what we show in Fig.1is the spatially varying part of (V,)(x): F(xxi), which is defined through (V,)(x) =(1/2pneo)(c/B)2ItyI2 (xi)F(x, x1 ). The rapid spatial variation of (V,)(x) occurs in a smallregion near the hybrid ion-ion resonance, which, in this case, is a little inside the minorityion resonance position RO. The strong spatial variation of the flow is due to the strongspatial variation of the plasma response S.. The width of the variation (shear layer width)7is approximately 2.5cm. Outside this region, the small and constant (shearless) flow is dueto the radial propagation of the wave. In Figure 2, we show the spatial profile of the flowshear d(V)/dx. The localization of the flow shear near the hybrid ion-ion resonance isevident. This figure also shows that the mean poloidal flow is not only strongly sheared,but also strongly curved (large d2 (V,)/dX2 ).As an application, we consider turbulence suppression by fast wave induced shearflow. The physics of shear flow suppression of turbulence has been discussed extensively inRef.3. The criterion is that the shearing rate of the poloidal flow exceeds the turbulencedecorrelation rate Awg: ke(Axkd(Vy,)/Ox) > Awg, where ke is the spectral averagedpoloidal wavenumber and Axk is the radial correlation length of the turbulence. For theparticular example given above, the flow shear can be inferred from Fig.2: d(Vy)/dx ~6.0 x 106 I,1 2 (xl)s-1. Assuming drift wave characteristic of edge turbulence, namely,kep, ~ 0.2, koAx ~ 1, Awg ~ w,* = 2.3 x 10 4 /s, and the plateau collisional regime forthe ions11 uneo = Iwti = 8.3 x 103/s, where p, is the ion Larmor radius at electrontemperature, w,* = (kop,)(c 8 /L.) is the electron diamagnetic drift frequency, c. is the ionsound speed, L = a is the plasma density scale length, and wti = v/JR is the ion transientfrequency, we obtain Ey 2.0kV/m for turbulence suppression. At this magnitude of theelectric field, we can estimate the total heating power absorbed by the plasma (RF powerthreshold). The total heating power Pt absorbed by the plasma (assuming minority ionresonance only) is given by9 : P", = 7r2aR2 (pm n)(c/B)2jI+j2, where E+ = E, + it, isthe left-hand component of fast wave electric field. Using Eq. (2) and the parameters givenabove, we obtain Pt 7.5kW, a number substantially smaller than those currently used inplasma heating experiments. The negligible heating effect at lower plasma temperaturesis an important feature of the fast wave flow drive. Such feature can be utilized to drive a8strongly sheared poloidal flow in the edge region of a tokamak, where less plasma heating isdesirable because it leads to less wall sputtering and thus less impurity influx. The physicalreason for the decoupling between the flow drive and power absorption at lower plasmatemperatures is that the fast wave flow drive is determined by the spatial variation of theplasma response across the minority ion resonance layer. At lower plasma temperatures,the minority ion resonance layer is narrower, the plasma response varies faster, the flowdrive is stronger, and thus less RF power is needed.In summary, we have shown that strongly sheared poloidal flow can be generated ina small region near the minority ion resonance using externally launched fast waves. Theflow drive is due to the spatial variation of the plasma response and depends strongly onlocal plasma parameters such as temperature and density. In particular, the flow drive isstronger at lower plasma temperatures (as in the edge region of a tokamak).AcknowledgementI like to thank Prof. P. H. Diamond for suggesting this problem and many usefuldiscussions. I also like to thank Prof. D. J. Sigmar for his interest and encouragement.This work was supported by the United States Department of Energy under Contract No.DE-FG02-91ER-54109 with the Massachusetts Institute of Technology.9References1. R. J. Groebner, K. H. Burrell, and R. P. Seraydarian, Phys. Rev. lett. 64, 3105(1990).2. Ch. P. Ritz, et al. Phys. Rev. Lett. 65, 2543 (1990).3. H. Biglari, P. H. Diamond, and P. W. Terry, Phys. Fluids B 2, 1 (1990).4. G. G. Craddock, and P. H. Diamond, Phys. Rev. Lett. 67, 1535 (1991).5. G. G. Craddock, P. H. Diamond, H. Biglari, M. Ono, and Y. L. Ho, Bull. Am. Phys.Soc. 36, 2411 (1991).6. S. J. Buchsbaum, Phys. Fluids 3, 418 (1960).7. F. W. Perkins, Nucl. Fusion 17, 1197 (1977).8. V. P. Bhatnagar, J. Jacquinot, D. F. H. Start, and B. J. D. Tubbing, Nucl. Fusion33, 83 (1993).9. T. H. Stix, Nucl. Fusion 15, (1975).10. T. H. Stix, Theory of Plasma Waves (McGraw-Hill, New York, 1962).11. S. P. Hirshman, Nucl. Fusion 18, 917 (1978).10Figure Captions1. Figure 1. Spatial profile of the flow velocity (VY,)(x). The minority ion resonancelocation is chosen at R0=190cm. Note the rapid spatial variation of the flow velocity(flow shear) near the hybrid ion-ion resonance, which is a little inside &b.2. Figure 2. Spatial profile of the flow shear d(V)(x), localized near the hybrid ion-ionresonance where the rapid spatial variation of plasma response occurs.11185.0 190.0 195.0R(cm)Figure 1by Gang120.100.060.02-0.02-0.06D00il:I * ** I *-0.10 118( 200.0.0185.0 190.0 195.0R(cm)Figure 2by Gang130.20.0-0.2-0.4-0.6cu0F-* p * * p3 * *-0.8 '18 200.0).0

Fast wave flow drive in a two-component plasma with low minority concentration

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Fast wave flow drive in a two-component plasma with low minority concentration

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